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Trees that can be grown in "too many" ways: A review of Bouch's construction

Hal Tasaki

TL;DR

The paper analyzes Bouch's hierarchical construction that yields rooted trees on the square lattice with a super-exponential number of growth orders. It shows that the number of growth sequences $N(T)$ can be bounded below by $N(T) \ge L!/C^{L}$ along an infinite subsequence of bond counts, by relating $N(T)$ to a Weight function $W(T)$ and bounding this weight through a carefully engineered tree sequence $T_j$. The key technical move is a parametrization by $E_j$ that converts the combinatorial growth into a tractable recursive bound on $W_j$, ultimately establishing $W_j \le C^{L_j}$ and hence $N(T_j) \ge L_j!/C^{L_j}$. This result has implications for operator growth in quantum spin systems and connects combinatorial tree growth to dynamical properties in higher dimensions.

Abstract

We carefully review the hierarchical construction by Bouch [Bouch2015] of trees on the square lattice that can be grown from its root in $L!/C^L$ distinct ways, where $L$ denotes the number of bonds constituting the tree, and $C>1$ is a constant. (As discussed in Section IV.A of [ParkerCaoAvdoshkinScaffidiAltman2019] and Appendix A.3 of [ShiraishiTasaki2024], this result has an implication on the operator growth in quantum spin systems in two or higher dimensions.)

Trees that can be grown in "too many" ways: A review of Bouch's construction

TL;DR

The paper analyzes Bouch's hierarchical construction that yields rooted trees on the square lattice with a super-exponential number of growth orders. It shows that the number of growth sequences can be bounded below by along an infinite subsequence of bond counts, by relating to a Weight function and bounding this weight through a carefully engineered tree sequence . The key technical move is a parametrization by that converts the combinatorial growth into a tractable recursive bound on , ultimately establishing and hence . This result has implications for operator growth in quantum spin systems and connects combinatorial tree growth to dynamical properties in higher dimensions.

Abstract

We carefully review the hierarchical construction by Bouch [Bouch2015] of trees on the square lattice that can be grown from its root in distinct ways, where denotes the number of bonds constituting the tree, and is a constant. (As discussed in Section IV.A of [ParkerCaoAvdoshkinScaffidiAltman2019] and Appendix A.3 of [ShiraishiTasaki2024], this result has an implication on the operator growth in quantum spin systems in two or higher dimensions.)

Paper Structure

This paper contains 7 sections, 2 theorems, 12 equations, 5 figures.

Table of Contents

  1. Introduction and the main theorem
  2. Proof
  3. Basic lemma and strategy
  4. The basic structure of Bouch's trees
  5. The recursion relation for the weights of the trees
  6. Descritpion in terms of $E_j$
  7. Final estimates

Key Result

Theorem 1.1

There are a constant $C>1$ and an infinite set $G$ of positive integers such that, for any $L\in G$, there exists a rooted tree on $\mathbb{Z}^2$ with $L$ bonds that satisfies e:main.

Figures (5)

  • Figure 1: Examples of rooted trees and the number of ways to construct them. See Figure \ref{['f:weights']} for the computation of $N(T)=210$ for the final example.
  • Figure 2: For simplicity, we consider the Bethe lattice with coordination number three. We first note that the number of ways to grow a tree with $L$ bonds from a fixed root is $3\times4\times5\times\cdots\times(L+2)$. To see this, note that when the growing tree has $n$ bonds, one can add the next bond to one of the $n+3$ positions at the boundary. The left figure depicts the case with $n=5$, where there are eight candidates shown by dotted lines. We next observe that the number of distinct trees with $L$ bonds, on the other hand, is upper bounded by $3^{2L}$. This is because any tree with $L$ bonds can be realized as a track of a random walk with $2L$ steps that starts at the root. See the right figure. We thus conclude that there exists a tree that can be grown in at least $3\times\ldots\times(L+2)/9^L>L!/9^L$ ways. Note that the proof is non-constructive.
  • Figure 3: The weights of bonds and trees in some examples. The number of ways to grow a tree, $N(T)$, is computed by using the formula \ref{['e:NW']}.
  • Figure 4: Schematic figures of the first three generations, $T_1$, $T_2$, and $T_2$, of Bauch's sequence of trees.
  • Figure 5: The tree $T_j$ consists of a horizontal segment with $\ell_j$ bonds and $b_j$ vertical branches sticking out of it. Each branch is a rotated copy of the tree $T_{j-1}$. We shall choose various parameters so that most of the bonds in $T_j$ belong to branches with the youngest generations when $j$ is large.

Theorems & Definitions (2)

  • Theorem 1.1: Bouch's theorem
  • Lemma 2.1